If you have the VZW version then you have the SCH-I545. The Google Play edition is purchased directly from Google via the Play store. The developer edition is also purchased right from Google. So, if you bought yours at the VZW then it is not the dev or GP edition, its the I545.

The easy way to verify the exact model of any phone is to go into settings, more, about phone and look at the model number field. The device name above it is identical by default but can be changed. The hardware version at the bottom shows the model, and also shows the hardware version.